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Introduction

In the Warm-Up, you compared the standard and factored forms of polynomial expressions. Both forms reveal different information about the relationship they model.

Let's take a look at the quadratic equation [tex]f(x) = x^2 - x - 6[/tex]. In this equation, the function is equal to a factorable polynomial expression. By finding the binomial factors of the resulting trinomial, we can rewrite the function as [tex]f(x) = (x - 3)(x + 2)[/tex].

Recall that if [tex](x - m)[/tex] is a factor of [tex]f[/tex], then [tex]m[/tex] is a zero of [tex]f[/tex]. So we can expect the graph of this equation to have zeros, and cross the [tex]x[/tex]-axis, at [tex]x = 3[/tex] and [tex]x = -2[/tex].

In this lesson, we'll use similar techniques to identify the zeros, or solutions, of higher-order polynomials and sketch their graphs.

Sagot :

GinnyAnswer
Sure, let's go through the detailed step-by-step solution for the quadratic equation [tex]\(f(x) = x^2 - x - 6\)[/tex].

### Step 1: Understanding the given quadratic equation

We start with the quadratic equation:
[tex]\[ f(x) = x^2 - x - 6 \][/tex]

### Step 2: Factoring the quadratic equation

To find the factors of [tex]\(x^2 - x - 6\)[/tex], we look for two binomials that multiply together to give the original quadratic expression. The quadratic expression can be factored into:
[tex]\[ f(x) = (x - 3)(x + 2) \][/tex]

### Step 3: Verification of the factors

To ensure correctness, we can expand the binomials:
[tex]\[ (x - 3)(x + 2) = x(x + 2) - 3(x + 2) \][/tex]
[tex]\[ = x^2 + 2x - 3x - 6 \][/tex]
[tex]\[ = x^2 - x - 6 \][/tex]

This confirms that our factored form is correct:
[tex]\[ f(x) = (x - 3)(x + 2) \][/tex]

### Step 4: Finding the zeros of the function

The zeros of [tex]\(f(x)\)[/tex] are the values of [tex]\(x\)[/tex] that make [tex]\(f(x) = 0\)[/tex].

Set the factored form equal to zero:
[tex]\[ (x - 3)(x + 2) = 0 \][/tex]

Using the zero-product property, we set each factor equal to zero:
[tex]\[ x - 3 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]

[tex]\[x + 2 = 0 \][/tex]
[tex]\[ x = -2 \][/tex]

### Step 5: Graphical interpretation

The solutions to the equation [tex]\( f(x) = 0 \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = -2 \)[/tex]. These are the points where the graph of the quadratic function crosses the x-axis.

### Summary

The quadratic function [tex]\( f(x) = x^2 - x - 6 \)[/tex] can be factored as [tex]\( (x - 3)(x + 2) \)[/tex]. The zeros of the function, where the graph intersects the x-axis, occur at [tex]\( x = 3 \)[/tex] and [tex]\( x = -2 \)[/tex].

Thus, we have the function in factorized form:
[tex]\[ f(x) = (x - 3)(x + 2) \][/tex]
And the zeros of the function are:
[tex]\[ x = 3 \][/tex]
[tex]\[ x = -2 \][/tex]

By following these steps, we successfully identified the factors and solutions of the quadratic equation.
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